Integrative and Comparative Biology
Integrative and Comparative Biology, volume 52, number 5, pp. 576–587
doi:10.1093/icb/ics096
Society for Integrative and Comparative Biology
SYMPOSIUM
Passive Robotic Models of Propulsion by the Bodies and Caudal
Fins of Fish
George V. Lauder,1,* Brooke Flammang* and Silas Alben†
*Museum of Comparative Zoology, Harvard University, Cambridge, MA 02138, USA; †School of Mathematics,
Georgia Institute of Technology, Atlanta, GA 30332, USA
From the symposium ‘‘Combining Experiments with Modeling and Computational Methods to Study Animal
Locomotion’’ presented at the annual meeting of the Society for Integrative and Comparative Biology, January 3–7, 2012
at Charleston, South Carolina.
1
E-mail: [email protected]
Synopsis Considerable progress in understanding the dynamics of fish locomotion has been made through studies of live
fishes and by analyzing locomotor kinematics, muscle activity, and fluid dynamics. Studies of live fishes are limited,
however, in their ability to control for parameters such as length, flexural stiffness, and kinematics. Keeping one of these
factors constant while altering others in a repeatable manner is typically not possible, and it is difficult to make critical
measurements such as locomotor forces and torques on live, freely-swimming fishes. In this article, we discuss the use of
simple robotic models of flexing fish bodies during self-propulsion. Flexible plastic foils were actuated at the leading edge
in a heave and/or pitch motion using a robotic flapping controller that allowed moving foils to swim at their
self-propelled speed. We report unexpected non-linear effects of changing the length and stiffness of the foil, and analyze
the effect of changing the shape of the trailing edge on self-propelled swimming speed and kinematics. We also quantify
the structure of the wake behind swimming foils with volumetric particle image velocimetry, and describe the effect of
flexible heterocercal and homocercal tail shapes on flow patterns in the wake. One key advantage of the considerable
degree of control afforded by robotic devices and the use of simplified geometries is the facilitation of mathematical
analyses and computational models, as illustrated by the application of an inviscid computational model to propulsion by
a flapping foil. This model, coupled with experimental data, demonstrates an interesting resonance phenomenon in which
swimming speed varies with foil length in an oscillatory manner. Small changes in length can have dramatic effects on
swimming speed, and this relationship changes with flexural stiffness of the swimming foil.
Introduction
The study of structure and function in different
species is a hallmark of comparative biology. Documenting how morphology and physiology vary both
among closely related species and across broader
phylogenetic scales is perhaps the most common
approach used in the field of functional morphology
and evolutionary biomechanics. This comparative
approach has both strong theoretical underpinnings
(Feder et al. 1987; Harvey and Pagel 1991; Garland
and Adolph 1994; Vogel 2003; Felsenstein 2004) and
a wealth of experimental techniques that can be utilized to better understand differences among species
in their structure and function (Wainwright et al.
1976; Biewener 1992).
The comparative approach, however, has significant limitations. Chief among these is the inability
to study the effect on performance of individual
traits while keeping all other aspects of morphology
and physiology constant. It is certainly possible to
reconstruct evolutionary patterns of morphology
and physiology, but many correlated changes occur
in animals’ structure and function that make isolating individual traits impossible. For example, one
noteworthy trend in the evolution of fishes is the
change in tail structure from the morphologically
asymmetrical heterocercal tail shape seen in sharks
and other basal ray-finned fishes to the externally
symmetrical homocercal tail of most teleost fishes
(Kardong 1998; Lauder 2000; Liem et al. 2001;
Advanced Access publication June 26, 2012
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Robotic models of fish locomotion
Lauder 2006). Using the comparative method, it is
not possible to isolate the effect of tail shape alone
on locomotor performance, while keeping all other
traits constant. Sharks differ from bony fishes in so
many ways that it is not particularly useful to compare species in these two groups if the specific goal is
to understand the effect of tail shape on swimming
performance.
Additional constraints on the comparative method
include the frequent lack of suitable variation among
extant species, so that certain hypotheses simply
cannot be tested. If we wished to study the effect
of body length on locomotor performance in zebrafish, we would not be able to conduct a comparative
study of live individuals over a size range that extends up to 1 m. Of course, many other changes in
zebrafish’s anatomy and physiology occur as body
length changes, so once again the limitations of comparing animals restrict our ability to isolate effects of
the trait of interest.
Two approaches that allow the isolation of
individual traits and that are increasingly available
to comparative biologists are (1) the construction
of robotic devices, and (2) the use of computational
models. Robotic devices provide a physical model
of animal function and allow individual traits to
be altered while controlling all others (Koditschek
et al. 2004; Koehl et al. 2011; Lauder et al. 2011a;
Long 2012). At the same time, many quantities of
interest in assessing locomotor performance such as
measurements of force and power output can be
made which are difficult to achieve on freely-moving
animals. In addition, computational modeling is a
powerful method that offers the ability to explore a
large parameter space and isolate individual traits
of interest (e.g., Mittal 2004; Alben 2009a, 2010;
Borazjani and Sotiropoulos 2010; Dong et al.
2012). For example, in a computational model of
swimming fishes it is relatively easy to alter body
stiffness and length, a manipulation that ranges
from difficult (Long et al. 1996) to impossible with
live animals.
In this article, our goals are (1) to use a simple
robotic model of swimming fishes to explore a
number of features of functional design in fishes,
including the effects of body length, stiffness, and
tail shape on locomotor performance, and (2) to
apply a computational model that incorporates key
features of the robotic system to explore in detail the
relationship between body length and swimming
performance. We show that even simple traits
such as body length or stiffness can have unexpected
non-linear effects on swimming, and we suggest
that complementary robotic and computational
approaches are an important adjunct to traditional
comparative methods for the study of organismal
function.
Robotic and computational methods
We have implemented a simple robotic model of the
bending body and traveling wave used during steady
rectilinear swimming by fishes. This robotic system,
discussed in more detail elsewhere (Lauder et al.
2007, 2011a, 2011b), uses flexible plastic foils (of
similar area and aspect ratio to moderate-sized
fishes) that are driven at their leading edge by a
computer-controlled robotic flapper. These foils
span the range of measured flexural stiffnesses of
passive fish bodies (McHenry et al. 1995; Long
1998; Long et al. 2002), and provide an excellent
model for body bending during undulatory locomotion by fish. The robotic mechanism controls both
heave and pitch motions of the foil’s leading edge,
and these movements generate a traveling wave that
produces forces and torques measured on the shaft
holding the flexible plastic foil.
These plastic foils swim in a recirculating flow
tank (with a cross-sectional area of 19 x 19 cm,
and a length of 50 cm parallel to the direction of
flow), and exhibit key characteristics of undulatory
locomotion in freely-swimming fishes (Lauder
and Tytell 2006): a wave of bending that travels
from anterior to posterior, Strouhal numbers that
range from 0.2 to 0.4, and Reynolds numbers of
10,000–50,000 (based on foil length). Tests of foil
swimming are thus done at-scale. These moving
foils are designed to reflect the thrust generating
region of swimming fishes, which for low to moderate swimming speeds is the posterior half of the body
in most species. Beat amplitudes of the foil tail are
thus not directly comparable to those of swimming
fishes relative to total length. Nonetheless, Strouhal
numbers of swimming foils and live fishes are
comparable, which indicates that the dynamics of
undulatory locomotion in fishes are well captured
by the swimming foil system (Triantafyllou and
Triantafyllou 1995).
The design of the robotic flapper includes a
low-friction air-carriage that allows the foil to selfpropel and to swim in place against a current that is
tuned to the mean swimming speed of the foil.
Changing the motion parameters, materials, and
shape of the foil results in changes in self-propelled
swimming speed, which are measured using a
custom LabView program. Studying self-propelling
objects is critical to understanding the dynamics of
locomotion, as emphasized elsewhere (Schultz and
578
Webb 2002; Lauder et al. 2007, 2011a; Borazjani and
Sotiropoulos 2010; Tytell et al. 2010). When swimming bodies self-propel, thrust and drag forces are
balanced over a tail beat cycle. If a swimming object
is towed at a different speed, thrust and drag forces
will not be balanced and the wake generated by the
swimming object can be quite different from that of
a self-propelled swimmer (Lauder et al. 2011a). One
advantage of studying the locomotion of flexible
plastic foils is that alterations in length and in the
shape of the trailing edge can be easily accomplished.
Furthermore, by changing the material in the foil it
is also a simple matter to alter the flexural stiffness of
the foil and to study the effect of stiffness on swimming performance.
In order to understand the mechanical and hydrodynamic basis of the foil’s swimming performance, it
is important to quantify both the three-dimensional
motion of the foil and the wake produced during
swimming. Three-dimensional movements of swimming foils were measured using three synchronized
Photron PCI-1024 digital video cameras, and a direct
linear transformation (DLT) calibration was implemented to provide accurate x, y, and z coordinates of
points on the swimming foils (Standen and Lauder
2005; Hedrick 2008; Flammang and Lauder 2009).
Volumetric measurement of the pattern of flow in
the foil’s wake was accomplished using the TSI V3V
volumetric flow visualization system (details provided in Flammang et al. [2011a, 2011b]).
G. V. Lauder et al.
Our computational modeling of foil swimming
used the inviscid analytical approach developed by
Alben et al. (2008, 2009b, 2012). This computational
approach is two-dimensional and uses measurements
of the foil’s length and flexural stiffness to compute
self-propelled swimming speeds and patterns of flow
in the wake while accounting for structure-fluid
interactions.
Results
Effects of length on swimming speed
One of the simplest possible parameters that could
be examined for effects on swimming performance is
length. How do alterations in the length alone of a
flexible swimming object affect propulsion? For studies of live fishes, one could examine changes in propulsion through ontogeny as fishes grow and
increase in length, but such comparisons are confounded by changes in mass, body shape, muscle
and respiratory physiology, for example, which all
change in conjunction with length as fishes grow.
A robotic approach allows study of the effects of
simple alterations in length of the foil on swimming
speed and provides a direct focus on the dynamics of
propulsion alone.
Figure 1 plots experimental data from the robotic
flapper that demonstrate unexpected non-linear effects of foil length on self-propelled swimming
speed. For example, the yellow plot (foil stiffness ¼
Fig. 1 Experimental data measuring self-propelled swimming speed versus foil length for flexible foils of three different flexural
stiffnesses. Foils were actuated with 1.0 cm heave amplitude at the leading edge at a frequency of 2 Hz, with no pitch motion.
Error bars of 1 SE are plotted for each point, but some error bars are obscured by the symbols. Note that certain stiffnesses of
foils (yellow plot) show multiple peaks in swimming speed as length of the foil changes, and that even relatively small changes in length
can have substantial effects on swimming speed. Foil flexural stiffnesses: coral color ¼ 4.9 103 Nm2; yellow ¼ 9.9 104 Nm2;
black ¼ 0.32 104 Nm2. (Reference to color applies to online version, not to printed black and white version.)
Robotic models of fish locomotion
9.9 104 Nm2) shows two distinct peaks and
troughs over the range of foil lengths studied. A
change of only 5 cm in length caused a decrease in
swimming speed of 33%, but a further increase in
foil length produced a 50% increase in swimming
speed. Swimming by foils of other stiffness also
showed significant non-linear changes in speed as
the length of the foil was increased: Prominent
peaks and valleys were evident in swimming speed
(Fig. 1). These experimental data demonstrate that
there is no simple relationship between swimming
speed and length or stiffness of the foils, and that
these two parameters interact in a complex manner.
Furthermore, the experimental data showing multiple peaks in swimming speed suggest that the flexible
foils are experiencing a resonance phenomenon in
which certain lengths interact with the moving
fluid in a manner that enhances propulsion, while
other lengths interact in a negative manner that decreases swimming speed.
To explore a broader range of lengths and more
precisely document this resonance phenomenon
than is possible from experimental data alone, we
used a two-dimensional inviscid analytical model of
swimming foils to calculate self-propelled swimming
speeds for a wide range of lengths and for foils of two
flexural stiffnesses (Fig. 2). The range of foil lengths
that we can study experimentally is limited by the size
of our flow tank, and small differences in
self-propelled speed are difficult to measure accurately. An inviscid computational approach can be
used to explore a much larger parameter space than
is possible experimentally. Comparison of the foil
shapes during swimming measured both experimentally and computed showed an excellent match (Fig.
2A and B). Furthermore, the computational model
revealed a distinct resonance pattern in which swimming speed showed multiple sharp peaks and valleys
as length increased, for foils of both studied stiffnesses
(Fig. 2C and D). These results confirm that even small
changes in length of foils can have dramatic effects on
swimming speed, and show that interpreting differences in swimming performance among flexible
self-propelling bodies requires knowledge of both
flexural stiffness and length of the object.
Effects of stiffness on swimming speed
Fish species vary in body stiffness and in how the
body is moved during swimming, but producing a
controlled range of variation in stiffness in a living
animal is challenging at best (see Long et al. 1996),
and it is thus difficult to understand the effect of
body stiffness on propulsion solely from comparisons
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Fig. 2 Results from an inviscid 2D numerical model of propulsion
of self-propelled flexible flapping foils (see Alben [2008, 2009b]
and Alben et al. 2012 for details of this model) which predicts
shape of the foil and the effects of changes in length of the foil on
swimming speed. Panels A and B compare experimentally
measured and computed midlines of two foils that differ in
stiffness, while panels C and D show plots of self-propelled
swimming speed versus length for two foils with different
stiffnesses. Each point is a separate simulation to calculate the
self-propelled swimming speed for that foil. Note the strong
resonance peaks that dramatically alter swimming speed as length
of the foil changes. Flexural stiffnesses: panel C foil material ¼
9.9 104 Nm2; panel D foil material ¼ 3.3 104 Nm2.
among swimming animals. A robotic approach that
involves swimming flexible foils of different stiffnesses and moving the leading edge with different
programs of motion allows us to investigate this
question in detail.
Figure 3 shows the results of experiments in which
self-propelled swimming speed is measured as foils of
different flexural stiffness are moved with two different programs of motion at their leading edge: a
heave only motion, and heave motion plus pitch
movements that impart additional energy into the
swimming foil. With the heave-only motion, a
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G. V. Lauder et al.
Fig. 3 Self-propelled speed versus flexural stiffness for flexible foils swimming in a recirculating flow tank. Each point represents a
different foil. Values plotted are 1 SE, but error bars are obscured by the symbols. Foils were 19 cm high, 6.8 cm long, and actuated at
the leading edge at 1 Hz with 1.5 cm heave, or 1.5 cm heave combined with 208 pitch motion. Foils actuated in heave only show a
distinct performance peak, but when pitch motion is added to heave actuation, a broad performance plateau is observed in which
swimming speed is relatively independent of flexural stiffness. From Lauder et al. (2011b).
Fig. 4 Propulsion by flexible foils with different trailing edge shapes. Foils have a flexural stiffness of 3.1 104 Nm2, and were
actuated at the leading edge with 1 cm heave at 2 Hz (the foil clamping rod is visible on the left side of the top row). Three
synchronized high-speed cameras captured lateral, posterior, and ventral views for the foils while they swam at their self-propelled
speed. White lines have been added to show the trailing edge in the middle row, and to show the ventral margin in the bottom
row (the foil 5 line extends to the tail fork). All foils are shown at the same relative time, when the leading edge is at maximum
lateral excursion. Details of foil area, relative shape, and self-propelled swimming speed are given in the text and in the caption
to Fig. 6.
Robotic models of fish locomotion
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distinct peak in performance was observed when
swimming speed was maximized at one value of flexural stiffness; but if pitch motion was added to the
heave actuation at the leading edge of the foil, this
peak disappeared and a broad plateau was observed
over which swimming performance was relatively insensitive to changes in flexural stiffness. These results
suggest that animals may be able to compensate for
changes in stiffness of the body (produced by activation of muscles or from growth and consequent
changes in body shape and skeletal features) by
changing the way in which propulsive waves are produced, thereby avoiding a decrease in performance at
higher stiffnesses. These data also show that the effects of foil stiffness on swimming performance very
much depend on how the foil is moved.
Effects of shape of the trailing edge on propulsion
Fish vary considerably in tail shape and yet even simple
comparisons among species include so many confounding factors that it is effectively impossible to isolate the effect of tail shape alone on swimming
performance. In a robotic flapping-foil system, however, it is easy to alter the shape of the trailing edge of
the foil to mimic the shapes of different fish tails and
explore the effect of tail shape alone on swimming.
Figure 4 shows the shapes taken by three foils
during swimming when powered by the robotic flapper. Foil 1 represents a morphologically symmetrical
homocercal fish tail with a vertical trailing edge, foil
Fig. 5 Midlines of foils as seen in ventral view for self-propelling
foils with different shapes of trailing edge. Details of foil area,
relative shape, and self-propelled swimming speed are given in
the text and in the caption to Fig. 6. Note that rectangular foil 1
displays two nodes which are also seen to a lesser extent in foil
5, but not in the foils with angled trailing edges.
Fig. 6 Self-propelled speeds (SPS) compared for foils (all of material flexural stiffness ¼ 3.1 104 Nm2) that differ in shape of the
trailing edge and in surface area. Foil-type numbers correspond to those described in Figs 4 and 5. Each foil was actuated at 1 cm
heave at the leading edge at 2 Hz frequency. Error bars are 2 SE. Foils were of different shapes and areas as follows: foil 1 ¼ square
trailing edge, area ¼ 131.2 cm2, dimensions ¼ 6.85 19.15 cm; foil 2 ¼ angled trailing edge, same length as foil 1, area ¼ 107.71 cm2;
foil 3 ¼ angled trailing edge, same area as foil 1; foil 4 ¼ angled trailing edge: same length and area as foil 1; foil 5 ¼ forked trailing
edge, same area as foil 1. From Lauder et al. (2011a). Statistical analysis of variation in swimming speed is given in the text.
582
3 has an angled trailing edge reminiscent of a shark’s
asymmetrical heterocercal tail, and foil 5 has a forked
shape with a notch centered between the dorsal and
ventral margins. All three of these foils have the same
surface area. Although the pattern of bending along
the length of each foil was generally similar, the distinct tail shapes did cause some differences in bending (Fig. 5): Foils 1 and 5 had two nodes along their
length, while the foils with angled trailing edges only
possessed a single node at the base of the tail.
Analysis of the self-propelled speeds of these foils
(and an additional one, foil 4, with the same length
and area as foil 1) demonstrates four key results
(Fig. 6). First, there is highly significant overall variation in swimming performance of these foils
(one-way analysis of variance F ¼ 46.9; P5.0001).
Second, the foil with the notched trailing edge (foil
5) had the worst swimming performance, and was
significantly slower than all foils except foil 1
(Tukey–Kramer Honestly Significant Difference post
hoc test). Third, the foils with the angled (heterocercal) trailing edge swam significantly faster than the
one with the symmetrical (homocercal) trailing edge,
but were not significantly different from each other.
When comparing two foils with the same surface
area (foils 1 and 3), the one with the heterocercal
shape swam 7% faster. However, as noted by
Lauder et al. (2011a), the foil with the heterocercal
shape requires more energy to swim at this speed,
and so the final costs of transport for the two foils
are roughly equivalent. Fourth, the foil with the
greatest surface area close to the actuation axis (foil
4) swims significantly faster than all other foils. This
result is concordant with previous data that indicate
that moving the material of the foil near the actuation axis while keeping total surface area constant
can more than double swimming speeds (Lauder
et al. 2011b)
Flexible flapping foils shed a vortex wake that can
be compared to the wake generated by freely swimming fishes. Figure 7 shows the three-dimensional
wake behind a flexible foil (like that of a bluegill
sunfish, Lepomis macrochirus) of the same flexural
stiffness as the foils shown in Fig. 6. This foil has a
more realistic fish-like tail shape, and it shed a vortex
wake that was similar in its general characteristics to
that shown for bluegill sunfish during steady swimming: a single large vortex ring formed by the caudal
fin and smaller vortical attachments produced by the
dorsal and anal fins (Tytell 2006; Flammang et al.
2011a). However, the strength of the vortical connections between the wakes of the dorsal fin and caudal
fin was much less than that in freely swimming bluegill sunfish (Tytell 2006; Flammang et al. 2011a).
G. V. Lauder et al.
Fig. 7 Visualization of water flow patterns in the wake of
the flexible foil shaped like a bluegill tail (panel A) when
swimming at its self-propelled speed (material flexural stiffness ¼ 3.1 104 Nm2). This shape (panel A) schematically
includes the trailing edges of the dorsal, anal, and caudal fins
(see [Drucker and Lauder 2001; Standen and Lauder 2005]).
Panel B shows patterns of flow in the wake imaged with volumetric particle image velocimetry (see text for discussion).
Vortical connections between the wakes of the caudal and dorsal
fins (arrow) are weak when compared to those observed in live
fish. Wake vorticity is isosurfaced and then colored by Y vorticity—red is high positive (6.5 s1), blue low (1.0 s1); X, Y, and Z
axes delimit the volume visualized, and units are in millimeters.
(Reference to color applies to online version, not to printed
black and white version.)
This may be due to the passive nature of the foil
as compared to those of bluegill sunfish which actively move and stiffen the dorsal and anal fins
during steady swimming (Jayne et al. 1996;
Drucker and Lauder 2001; Standen and Lauder
2005). This demonstrates that fin shape alone is
not a determinant of swimming hydrodynamics
and that active control of fin stiffness is an important
component of thrust direction and magnitude.
Analysis of the patterns of flow in
three-dimensional wakes produced by flexible foils
Robotic models of fish locomotion
583
Fig. 8 Visualization of patterns of flow in the wake behind self-propelling flexible foils (with material flexural stiffness ¼
3.1 104 Nm2) (panels A and D). These foils have an angled trailing edge that reflects the heterocercal tails of sharks either
schematically (panel A) or with a somewhat more realistic shape (panel D). These foils produce a single large vortex ring, but also a
weak secondary ring of lower vorticity that joins the upper portion of the large ring (panels B–E). At the values of vorticity isosurface
chosen, the small weaker upper ring may not be completely closed as seen in panels B and C. Vorticity of the wake is isosurfaced and
then colored by Y vorticity. Modified from Flammang et al. (2011b). (Reference to color applies to online version, not to printed black
and white version.)
with angled trailing edges (Fig. 8) reveals wakes that
are generally similar to those generated by swimming
sharks (Wilga and Lauder 2002, 2004; Flammang
et al. 2011b) with a single, large, inclined vortex
ring. The secondary ring observed by Flammang
et al. (2011b), however, was not observed in the
wakes and instead a weak, smaller, dorsally-located
ring is attached to the large vortex ring (Fig. 8).
These results suggest that passive foils with inclined
trailing edges are not capable of generating patterns
of flow in the wake that closely mimic those of
freely-swimming sharks. Instead, the highly flexible
dorsal tip of the angled foils seems to generate a
weak vortex ring near the top of the larger ring.
Three-dimensional analyses of the motions of
foils, and particularly of the trailing edge, can assist
in understanding the formation of the wake and
some of its key characteristics. As seen from behind
(posterior view), the lateral excursions of the symmetrical dorsal and ventral tips of tail foil 1 are
equivalent at 28.0 and 28.3 mm, respectively (Fig.
9A). The asymmetrical heterocercal shape (foil 3)
produced obvious asymmetries in lateral motion
where the ventral tail tip only moved through a zexcursion of 21.8 mm, while the dorsal tip moved
50% more (42 mm; Fig. 9B) than did the homologous location on symmetrical foil 1 (Fig. 9A). This
suggests that the relatively thin and pointed dorsal
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G. V. Lauder et al.
lobe of the heterocercal foil is passively flexing
during lateral motion and that this motion results
in larger excursions. Motion of the dorsal and ventral tail tips of the notched foil (foil 5; Fig. 9C) both
underwent greater lateral excursions (mean of
37.7 mm) than did the homologous points on symmetrical foil 1, again indicating that thin passive tail
regions will undergo greater lateral excursions than
do homologous regions of a symmetrical tail. The
center of the forked tail moves much less, with a
mean excursion of 17.8 mm.
Motion of the foil viewed from lateral and ventral
perspectives (Fig. 10) showed that the tips move in a
figure-eight pattern in the z and x dimensions. The
dorsal and ventral tail tips of symmetrical foil 1 only
moved 4.35 mm in the x-direction on average, while
the x-excursions of the dorsal tips of foils 3 and 5
were larger, 6.5 and 5.2 mm, respectively, indicative
of passive bending (Fig. 10). The dorsal and ventral
tips of forked foil 5 also moved to a greater extent
than did symmetrical foil 1 in the x-dimension
(mean ¼ 5.6 mm).
Discussion and synthesis
Fig. 9 Kinematics of self-propelled swimming foils (material
flexural stiffness ¼3.1 104 Nm2). Foil numbers correspond to
those shown in Fig. 6. Plots show the YZ plane from posterior
view, Fig. 4); point 1 is at the tip of the dorsal trailing edge, point
2 at the tip of the ventral trailing edge, and point 5 is located at
the fork of foil 5. Foil 1 has similar excursions (z-axis ranges) of
the dorsal and ventral tips of the tail, while the ventral margin of
foil 3 undergoes much smaller excursions and both dorsal and
ventral tips of this foil move in a curved trajectory, with a concave up and down motion, respectively. Dorsal and ventral tips of
foil 5 show greater lateral excursions than do the homologous
points on foil 1, while the point in the tail fork moves only half as
much as the dorsal and ventral tips.
This article has concentrated on passive flexible foils
activated only at the leading edge as a model system
for understanding aquatic propulsion. These passive
foils have proven to be excellent models for studying
locomotion of fish as they generate undulatory waves
that produce thrust and self-propel at Strouhal and
Reynolds numbers matching that of freely-swimming, similarly sized fish. The experimental and
computational data presented here show unexpected
non-linear effects of changing the length and stiffness
of swimming foils, results that could not have been
obtained from live fish. Additionally, the shape of
the trailing edge can have significant effects on swimming speed, and analysis of tail kinematics shows
that foils with an angled trailing edge swim faster.
The foil with the forked tail shape exhibits the slowest swimming speeds.
However, we have also presented a number of situations in which the motion of these passive foils
does not produce wake-flow patterns or kinematics
that closely match those previously observed for live
fishes. We suggest that these differences arise from
the ability of fishes to actively stiffen the tail region
using intrinsic musculature that is not represented in
the passive foils. For example, Flammang (2010) recorded muscle activity from the radialis muscle, intrinsic to the heterocercal caudal fin of sharks. These
data, summarized in Fig. 11, show that the radialis
muscle is electrically active at the point during the
Robotic models of fish locomotion
585
Fig. 10 Kinematics of self-propelled swimming foils (material flexural stiffness ¼ 3.1 104 Nm2). Foil numbers correspond to those
shown in Fig. 6. Plots show the lateral (YX) and ventral (ZX) views, Fig. 4; point 1 is at the dorsal tail tip, point 2 at the ventral tail tip,
and point 5 is located at the fork of foil 5. The points on the tips of all foils move in a figure-eight pattern, and are passively bent against
oncoming flow. This is reflected in the lateral excursions of the tips of points 1 and 2. These excursions are greater for foils 3 and 5
compared to foil 1 which has a symmetrical shape (see text for further discussion).
Fig. 11 Evidence of active stiffening in the tail of a
freely-swimming adult spiny dogfish (Squalus acanthias). Plots
schematically show the tail beat from one side to the other
during steady swimming (solid black line), tail-beat velocity
(dotted line), and drag force on the tail (large dashed line)
under the simple assumption that drag is proportional to
tail-velocity squared. Colored bars show measured electrical
activity in the radialis muscle (intrinsic to the tail, with two
electrodes in each muscle) on the right (blue) and left (red) sides.
The timing of activity of the radialis muscle suggests that it acts to
stiffen the tail when imposed fluid forces are highest. Error
bars are 1 SE. (Reference to color applies to online version, not
to printed black and white version.)
tail beat when fluid loading on the fin is expected to
be highest, and that there is considerable overlap
between muscles from the right and left sides that
could also stiffen the caudal fin.
Similar results were obtained for bluegill sunfish
(Flammang and Lauder 2008) in which recordings of
intrinsic tail muscles during steady swimming
showed that these muscles actively stiffen the
caudal fin. During braking and other types of maneuvering, intrinsic caudal fin musculature also controls conformation of the tail (Flammang and Lauder
2009). The extent of active control can be seen in
Fig. 12, which shows a bluegill sunfish executing a
braking maneuver. The dorsal and ventral lobes of
the tail move to opposite sides of the body. Such
control is not possible in passive foils.
Robotic models that possess active stiffening
against imposed flow (Phelan et al. 2010; Tangorra
et al. 2011; Esposito et al. 2012) demonstrate that the
cupping behavior frequently observed in the moving
fins of live fish can generate increased thrust (Tytell
2006; Lauder et al. 2007; Lauder and Madden 2007).
If the passive model foils studied here could be enhanced with active stiffening elements so that
the thin flexible tips were able to lead the tail in
lateral motion, then we would predict enhanced
thrust and increased swimming speeds. We would
586
G. V. Lauder et al.
References
Fig. 12 Posterior view of a bluegill sunfish (L. macrochirus) executing a braking maneuver. This image shows active control of
conformation of the tail with the dorsal lobe (red color) having
moved to the upper left, and the ventral lobe (blue color) moving
to the lower right. Such active control features of fish tails are
not well modeled by passive foils. (Reference to color applies to
online version, not to printed black and white version.)
also expect that such actively stiffened models would
produce patterns of flow in the wake with more
robust connections between the wakes of the
dorsal, anal, and caudal fins, and that wakes of
angled foils would more closely resemble those observed in live sharks.
Incorporation of the capability for stiffening that
can be activated at precise phases and forces is a key
challenge for the future design of simple robotic devices simulating undulatory locomotion in fishes.
Acknowledgments
The authors thank Vern Baker and Chuck Witt for
their invaluable assistance with this research, and the
James Tangorra Laboratory for measuring flexural
stiffness of foils. Tyson Strand and Dan Troolin
from TSI Corporation kindly provided the V3V
equipment used to quantify volumetric wake flows.
Funding
The
National
Science
Foundation
grants
(EFRI-0938043 to G.V.L. and NSF-DMS 0810602
and 1022619 to S.A.).
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